Join an edge in a mesh
That edge is joined to the mesh, but there are three ways that it might look the way it does.

1) It's not part of a FACE. To make it part of a face, select the (at least 3) vertices that you want it to be part of, and press F. That will make a polygon out of those vertices. If you've chosen correctly, that edge will be part of the polygon as well

2) It's a 2-sided polygon. This is a really strange case, but the solution is usually just to delete it. You can switch to Edge Select mode, select that edge and delete it. You might have to do that multiple times to delete all of the edges that are on top of each other.

3) It IS part of a face, but you're looking at the backside of the face. Polygons only have one face. This is defined by the direction that the normal is facing for that polygon. The normal is a vector that points out from the surface of the face, perfectly perpendicular to it. If you're looking at the polygon from the same side as the normal, then you can see it. If you're looking from the other side, the polygon will disappear, and seem transparent. To solve this problem, confirm that it's a face and which direction it's pointing. In the viewport, TAB into edit mode. Then press N to open the view properties pane if it's not already open. Somewhere in that pane, you'll see a trio of buttons. One of them will be a box with a face highlighted. Click that box to turn on "Draw Face Normals." This will draw a little (or not so little, if the size is set too high) blue line that indicates the normal. The line starts ON the face of the polygon, and points outward. Look at the polygon from the same side as the normal (the normal should be pointing more-or-less at you) and the face should be visible.

Hope that helps!